These slides illustrate a few example R commands that can be useful for the analysis of repeated measures data.

Transcription

1 These slides illustrate a few example R commands that can be useful for the analysis of repeated measures data. We focus on the experiment designed to compare the effectiveness of three strength training programs. 1

2 #Read the data d=read.delim(" ~dnett/s511/repeatedmeasures.txt") head(d) Program Subj Time Strength

3 #Create factors d$program=as.factor(d$program) d$subj=as.factor(d$subj) d$timef=as.factor(d$time) head(d) Program Subj Time Strength Timef

4 # Compute sample means means = tapply(d$strength,list(d$time,d$program),mean) means

5 # Make a profile plot of the means x.axis = unique(d$time) par(fin=c(6.0,6.0),pch=18,mkh=.1,mex=1.5, cex=1.2,lwd=3) matplot(c(2,14), c(79,85.7), type="n", xlab="time(days)", ylab="strength", main= "Observed Strength Means") matlines(x.axis,means,type='l',lty=c(1,2,3)) matpoints(x.axis,means, pch=c(16,17,15)) legend(2.1,85.69,legend=c("ri program", 'WI Program','Controls'), lty=c(1,2,3),col=1:3,bty='n') 5

6 6

7 The following code illustrates how to specify different models for the variance-covariance of the response vector. We will begin with each model specification followed by a description of the model variancecovariance matrix associated with that model specification. Then output from each model fit will be examined. At first, we will assume the same structure for the mean in each case (one mean for each combination of program and time, a cell means model). Later we will look into different models for the mean. 7

8 This code fits a linear mixed effects model with independent random effects for each subject. The resulting variance-covariance structure for the response vector is block diagonal. Each block has a compound symmetric structure. There is one block for each subject. lme(strength ~ Program*Timef,data=d, random= ~ 1 Subj) 8

9 Var(y) is block diagonal with blocks σe 2 + σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σe 2 + σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σe 2 + σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σe 2 + σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σe 2 + σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σe 2 + σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σs 2 σe 2 + σs 2 9

10 This code fits a general linear model. The variancecovariance structure for the response vector is block diagonal. Each block has a compound symmetric structure. There is one block for each subject. This code fits a model for the response vector that is identical to the model obtained using the previous lme code. gls(strength ~ Program*Timef,data=d, correlation = corcompsymm(form=~1 Subj)) 10

11 Var(y) is block diagonal with blocks 1 ρ ρ ρ ρ ρ ρ ρ 1 ρ ρ ρ ρ ρ ρ ρ 1 ρ ρ ρ ρ σ 2 ρ ρ ρ 1 ρ ρ ρ ρ ρ ρ ρ 1 ρ ρ ρ ρ ρ ρ ρ 1 ρ ρ ρ ρ ρ ρ ρ 1 To match with previous variance components, note that σ 2 = σ 2 e + σ 2 s σ 2 ρ = σ 2 s ρ = σ2 s σ 2 = σ2 s. σe 2 + σs 2 11

12 This code fits a general linear model. The variancecovariance structure for the response vector is block diagonal. Each block has an AR(1) structure. There is one block for each subject. gls(strength ~ Program*Timef,data=d, correlation = corar1(form=~1 Subj)) 12

13 Var(y) is block diagonal with blocks 1 ρ ρ 2 ρ 3 ρ 4 ρ 5 ρ 6 ρ 1 ρ ρ 2 ρ 3 ρ 4 ρ 5 ρ 2 ρ 1 ρ ρ 2 ρ 3 ρ 4 σ 2 ρ 3 ρ 2 ρ 1 ρ ρ 2 ρ 3 ρ 4 ρ 3 ρ 2 ρ 1 ρ ρ 2 ρ 5 ρ 4 ρ 3 ρ 2 ρ 1 ρ ρ 6 ρ 5 ρ 4 ρ 3 ρ 2 ρ 1 13

14 This code fits a general linear model. The variancecovariance structure for the response vector is block diagonal. Each block is a general symmetric, positive definite variance-covariance matrix. There is one block for each subject. gls(strength ~ Program*Timef,data=d, correlation = corsymm(form=~1 Subj), weight = varident(form = ~ 1 Timef)) 14

15 Var(y) is block diagonal with blocks 1 σ 2 diag(δ 1,..., δ 7 ) 1 ρ ij ρ ij 1 1 diag(δ 1,..., δ 7 ) 15

16 = σ 2 δ 2 1 δ2 2 δ3 2 δ4 2 δ 2 5 ρ ij δ i δ j ρ ij δ i δ j δ 2 6 Identifiability Constraint : δ 1 1 δ

17 To understand the reason for an identifiability constraint, notice that an arbitrary positive definite 7 7 covariance matrix depends on only = 7(7 + 1) 2 = 28 parameters. However, we have σ 2, = 21 ρ ij parameters, and δ 1,..., δ 7. That s 29 parameters for a symmetric positive definite matrix that depends on at most 28 parameters. 17

18 Thus, R chooses to set δ 1 to 1. Without such a constraint, it is easy to use different values of the parameters to define the same matrix. For example, [ ] [ ] 1 1 = 3 3 = σ δ δ 2 7 ρ [ ] 18

19 If you are interested in learning about how to fit other variance-covariance structures in R, the following help commands will be useful.?corclasses?varclasses?pdclasses 19

20 # Use the lme function. This application # assumes that each subject has a different # identification value library(nlme) d.lme = lme(strength ~ Program*Timef, random= ~ 1 Subj, data=d, method="reml") summary(d.lme) Linear mixed-effects model fit by REML Data: d AIC BIC loglik Random effects: Formula: ~1 Subj (Intercept) Residual StdDev:

21 Fixed effects: Strength ~ Program * Timef Value Std.Error DF t-value p-value (Intercept) Program Program Timef Timef Timef Timef Timef Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef

22 Number of Observations: 399 Number of Groups: 57 anova(d.lme) numdf dendf F-value p-value (Intercept) <.0001 Program Timef <.0001 Program:Timef

23 # Use the gls( ) function to fit a # model where the errors have a # compound symmetry covariance structure # within subjects. Random effects are # not used to induce correlation. d.glscs = gls(strength ~ Program*Timef,data=d, correlation = corcompsymm(form=~1 Subj), method="reml") summary(d.glscs) Generalized least squares fit by REML Model: Strength ~ Program * Timef Data: d AIC BIC loglik

24 Correlation Structure: Compound symmetry Formula: ~1 Subj Parameter estimate(s): Rho Coefficients: Value Std.Error t-value p-value (Intercept) Program Program Timef Timef Timef Timef Timef Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef

25 Program2:Timef Program3:Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef Residual standard error: Degrees of freedom: 399 total; 378 residual anova(d.glscs) Denom. DF: 378 numdf F-value p-value (Intercept) <.0001 Program Timef <.0001 Program:Timef

26 # Try an auto regressive covariance # structures across time within # subjects d.glsar = gls(strength ~ Program*Timef,data=d, correlation = corar1(form=~1 Subj), method="reml") summary(d.glsar) Generalized least squares fit by REML Model: Strength ~ Program * Timef Data: d AIC BIC loglik Correlation Structure: AR(1) Formula: ~1 Subj Parameter estimate(s): Phi

27 Coefficients: Value Std.Error t-value p-value (Intercept) Program Program Timef Timef Timef Timef Timef Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef

28 Program2:Timef Program3:Timef Residual standard error: Degrees of freedom: 399 total; 378 residual anova(d.glsar) Denom. DF: 378 numdf F-value p-value (Intercept) <.0001 Program Timef Program:Timef

29 # Use an arbitray covariance matrix for # observations at different time # points within subjects d.gls = gls(strength ~ Program*Timef,data=d, correlation = corsymm(form=~1 Subj), weight = varident(form = ~ 1 Timef), method="reml") summary(d.gls) Generalized least squares fit by REML Model: Strength ~ Program * Timef Data: d AIC BIC loglik

30 Correlation Structure: General Formula: ~1 Subj Parameter estimate(s): Correlation:

31 Variance function: Structure: Different standard deviations per stratum Formula: ~1 Timef Parameter estimates:

32 Coefficients: Value Std.Error t-value p-value (Intercept) Program Program Timef Timef Timef Timef Timef Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef Program2:Timef Program3:Timef

33 Program2:Timef Program3:Timef Residual standard error: Degrees of freedom: 399 total; 378 residual anova(d.gls) Denom. DF: 378 numdf F-value p-value (Intercept) <.0001 Program Timef <.0001 Program:Timef

34 # Compare the fit of various covariance # structures. anova(d.gls, d.glscs) Model df AIC BIC loglik Test L.Ratio p-value d.gls d.glscs vs <.0001 anova(d.gls, d.glsar) Model df AIC BIC loglik Test L.Ratio p-value d.gls d.glsar vs

35 # Treat time as a continuous variable and # fit quadratic trends in strength # over time d.time = gls(strength ~ Program+Time+ Program*Time+I(Time^2)+Program*I(Time^2), data=d, correlation = corar1(form=~1 Subj), method="reml") summary(d.time) Generalized least squares fit by REML Model: Strength ~ Program + Time + Program * Time + I(Time^2) + Program * I(Time^2) Data: d AIC BIC loglik

36 Correlation Structure: AR(1) Formula: ~1 Subj Parameter estimate(s): Phi Coefficients: Value Std.Error t-value p-value (Intercept) Program Program Time I(Time^2) Program2:Time Program3:Time Program2:I(Time^2) Program3:I(Time^2)

37 Residual standard error: Degrees of freedom: 399 total; 390 residual anova(d.time) Denom. DF: 390 numdf F-value p-value (Intercept) <.0001 Program Time I(Time^2) Program:Time Program:I(Time^2)

38 # To compare the continuous time model to the # model where we fit a different mean at each # time point, we must compare likelihood values # instead of REML likelihood values. d.glsarmle = gls(strength ~ Program*Timef, data=d, correlation = corar1(form=~1 Subj), method="ml") d.timemle = gls(strength ~ Program+ Time+ Program*Time+I(Time^2)+Program*I(Time^2), data=d, correlation = corar1(form=~1 Subj), method="ml") anova(d.glsarmle, d.timemle) Model df AIC BIC loglik Test L.Ratio p-value d.glsarmle d.timemle vs

39 # Do not fit different quadratic trends # for different programs d.timemle = gls(strength ~ Program+Time+ Program*Time+I(Time^2), data=d, correlation = corar1(form=~1 Subj), method="ml") anova(d.glsarmle, d.timemle) Model df AIC BIC loglik Test L.Ratio p-value d.glsarmle d.timemle vs

40 # Fit a model with random regression coefficients # for individual subjects d.timer = lme(strength ~ Program+Time+ Program*Time+I(Time^2), random = ~ Time + I(Time^2) Subj, data=d, correlation = corar1(form=~1 Subj), control=list(msmaxiter=100), method="reml") 40

41 d.timer Linear mixed-effects model fit by REML Data: d Log-restricted-likelihood: Fixed: Strength ~ Program + Time + Program * Time + I(Time^2) (Intercept) Program2 Program3 Time I(Time^2) Program2:Time Program3:Time Random effects: Formula: ~Time + I(Time^2) Subj Structure: General positive-definite, Log-Cholesky parametrization StdDev Corr (Intercept) (Intr) Time Time I(Time^2) Residual

42 Correlation Structure: AR(1) Formula: ~1 Subj Parameter estimate(s): Phi Number of Observations: 399 Number of Groups: 57 fixef(d.timer) (Intercept) Program2 Program3 Time I(Time^2) Program2:Time Program3:Time ranef(d.timer) (Intercept) Time I(Time^2) e e e e e-04 42

43 e e e e e e e e e e e e e e e e e e e e e-04 43

44 e e e e e e e e e e e e e e e e e e e e e-03 44

45 e e e e e e e e e e-03 coef(d.timer) (Intercept) Program2 Program3 Time I(Time^2) Program2:Time

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47 Program3:Time

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50 50

51 # Do we need the AR(1) structure in the # random coefficients model? d.timeru = lme(strength ~ Program+Time+ Program*Time+I(Time^2), random = ~ Time + I(Time^2) Subj, data=d, method="reml") anova(d.timer,d.timeru) Model df AIC BIC loglik Test L.Ratio p-value d.timer d.timeru vs e-04 # The more complicated model is preferred. # Keep the AR(1) structure. 51